   Chapter 13.2, Problem 6E

Chapter
Section
Textbook Problem

# (a) Sketch the plane curve with the given vector equation.(b) Find r'(t).(c) Sketch the position vector r(t) and the tangent vector r'(t) for the given value of t.6. r(t) = et i + 2t j, t = 0

(a)

To determine

To sketch: The plane curve with the vector equation r(t)=eti+2tj,t=0.

Explanation

Write the x-component of the vector r(t)=eti+2tj,t=0.

x=et

Take natural log on both sides of the expression.

ln(x)=ln(et)

Rearrange the expression ln(x)=ln(e2t) as follows.

ln(x)=tt=ln(x)

Write the y-component of the vector r(t)=eti+2tj,t=0.

y=2t (1)

Substitute ln(x) for t in equation (1),

y=2ln(x)

The required curve is a graph of y=2ln(x).

Find the vector r(t) at t=0

(b)

To determine

To find: The vector r(t) from the vector function r(t)=eti+2tj,t=0.

(c)

To determine

To sketch: The position vector r(t) and the tangent vector r(t) at t=0.

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